# zbMATH — the first resource for mathematics

Structure of a Morse form foliation on a closed surface in terms of genus. (English) Zbl 1223.57022
Let $$\omega$$ be a closed $$1$$-form with Morse singularities on a genus $$g$$ closed surface $$M$$. The form $$\omega$$ defines a foliation $$\mathcal{F} _{\omega}$$ on $$M\backslash Sing\;\omega$$ and a singular foliation $$\overline{\mathcal{F}}_{\omega}$$ on the whole $$M$$. A leaf $$\gamma \in\mathcal{F}_{\omega}$$ is compactifiable if $$\gamma\cup Sing\;\omega$$ is compact. Non-compactifiable leaves of $$\mathcal{F}_{\omega}$$ form minimal components $$\mathcal{C}_{j}^{\min};$$ each such leaf is dense in its minimal component. If $$\gamma$$ is a compact singular leaf, a small closed tubular neighborhood of $$\gamma$$ is denoted by $$V(\gamma)$$.
The main result of this work is the relation $$c(\omega)+\sum_{\gamma}g(V(\gamma))+g(\bigcup \overline{\mathcal{C}_{j}^{\min}})=g,$$ where $$c(\omega)$$ is the number of homologically independent compact leaves. This result allows the author to prove a criterion for compactness of $$\overline{\mathcal{F}}_{\omega}$$, to estimate the number of its minimal components and to give an upper bound on the rank $$rk\;\omega,$$ in terms of genus.

##### MSC:
 57R30 Foliations in differential topology; geometric theory 58K65 Topological invariants on manifolds
Full Text:
##### References:
  Aranson, S.K.; Zhuzhoma, E.V., On the structure of quasiminimal sets of foliations on surfaces, Russ. acad. sci. sb. math., 82, 397-424, (1995)  Arnoux, P.; Levitt, G., Sur lʼunique ergodicité des 1-formes fermées singulières, Invent. math., 84, 141-156, (1986) · Zbl 0577.58021  Farber, M., Topology of closed one-forms, Math. surv., vol. 108, (2004), AMS · Zbl 1052.58016  Gelbukh, I., Presence of minimal components in a Morse form foliation, Differ. geom. appl., 22, 189-198, (2005) · Zbl 1070.57016  Gelbukh, I., Number of minimal components and homologically independent compact leaves for a Morse form foliation, Stud. sci. math. hung., 46, 547-557, (2009) · Zbl 1274.57005  Gelbukh, I., On the structure of a Morse form foliation, Czech. math. J., 59, 207-220, (2009) · Zbl 1224.57010  I. Gelbukh, The number of minimal components and homologically independent compact leaves of a weakly generic Morse form on a closed surface, Rocky Mt. J. Math., in press. · Zbl 1280.57021  Harary, F., Graph theory, (1994), Addison-Wesley Publ. Comp. Reading, MA · Zbl 0797.05064  Imanishi, H., On codimension one foliations defined by closed one forms with singularities, J. math. Kyoto univ., 19, 285-291, (1979) · Zbl 0417.57010  Jiménez López, V.; Soler López, G., Transitive flows on manifolds, Rev. mat. iberoamericana, 20, 107-130, (2004) · Zbl 1063.54032  Kono, S., The structure of quasiminimal sets, Proc. Japan acad., 46, 599-604, (1970) · Zbl 0219.54041  Levitt, G., Feuilletages des surfaces, Ann. inst. Fourier, 32, 179-217, (1982) · Zbl 0454.57015  Levitt, G., 1-formes fermées singulières et groupe fondamental, Invent. math., 88, 635-667, (1987) · Zbl 0594.57014  Maier, A.G., Trajectories on closed orientable surfaces, Math. sb., 12, 71-84, (1943) · Zbl 0063.03856  Melʼnikova, I., An indicator of the noncompactness of a foliation on $$M_g^2$$, Math. notes, 53, 356-358, (1993)  Melʼnikova, I., A test for non-compactness of the foliation of a Morse form, Russ. math. surveys, 50, 444-445, (1995) · Zbl 0859.58005  Melʼnikova, I., Maximal isotropic subspaces of skew-symmetric bilinear mapping, Mosc. univ. math. bull., 54, 1-3, (1999) · Zbl 0957.57018  Zorich, A., Hamiltonian flows on multivalued Hamiltonians on closed orientable surfaces, (1994), Max-Planck Institut für Mathematik Bonn
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.